Nas–Walras equilibria of a large economy

Individuals exchange contracts for the delivery of commodities in competitive markets and, simultaneously, act strategically; actions affect utilities across individuals directly or through the payoffs of contracts. This encompasses economies with asymmetric information. Nash–Walras equilibria exist for large economies, even if utility functions are not quasi-concave and choice sets are not convex, which is the case in standard settings; the separation of the purchase from the sale of contracts and the pooling of the deliveries on contracts guarantee that the markets for commodities clear.

Photochemically induced nuclear spin polarization in reaction centers of photosystem II observed by 13C-solid-state NMR reveals a strongly asymmetric electronic structure of the P680.+ primary donor chlorophyll

We report 13C magic angle spinning NMR observation of photochemically induced dynamic nuclear spin polarization (photo- CIDNP) in the reaction center (RC) of photosystem II (PS2). The light-enhanced NMR signals of the natural abundance 13C provide information on the electronic structure of the primary electron donor P680 (chlorophyll a molecules absorbing around 680 nm) and on the pz spin density pattern in its oxidized form, P680⨥. Most centerband signals can be attributed to a single chlorophyll a (Chl a) cofactor that has little interaction with other pigments. The chemical shift anisotropy of the most intense signals is characteristic for aromatic carbon atoms. The data reveal a pronounced asymmetry of the electronic spin density distribution within the P680⨥. PS2 shows only a single broad and intense emissive signal, which is assigned to both the C-10 and C-15 methine carbon atoms. The spin density appears shifted toward ring III. This shift is remarkable, because, for monomeric Chl a radical cations in solution, the region of highest spin density is around ring II. It leads to a first hypothesis as to how the planet can provide itself with the chemical potential to split water and generate an oxygen atmosphere using the Chl a macroaromatic cycle. A local electrostatic field close to ring III can polarize the electronic charge and associated spin density and increase the redox potential of P680 by stabilizing the highest occupied molecular orbital, without a major change of color. This field could be produced, e.g., by protonation of the keto group of ring V. Finally, the radical cation electronic structure in PS2 is different from that in the bacterial RC, which shows at least four emissive centerbands, indicating a symmetric spin density distribution over the entire bacteriochlorophyll macrocycle.